the vanishing procyclicality of labor productivity
TRANSCRIPT
The Vanishing Procyclicality of Labor Productivity�
Jordi Galí
CREI, Universitat Pompeu Fabra and Barcelona GSE
Thijs van Rens
CREI, Universitat Pompeu Fabra and Barcelona GSE
This version: July 2010
First version: August 2008
Abstract
We document three changes in postwar US macroeconomic dynamics: (i) the
procyclicality of labor productivity has vanished, (ii) the relative volatility of em-
ployment has risen, and (iii) the relative (and absolute) volatility of the real wage
has risen. We propose an explanation for all three changes that is based on a com-
mon source: a decline in labor market frictions. We develop a simple model with
labor market frictions, variable e¤ort, and endogenous wage rigidities to illustrate
the mechanisms underlying our explanation. We show that the reduction in frictions
may also have contributed to the observed decline in output volatility.
Keywords: labor hoarding, labor market frictions, wage rigidities, e¤ort choice
JEL classi�cation: E24 E32
�We have bene�ted from comments and suggestions by Almut Balleer, Paula Bustos, Vasco Carvalho,Wouter Denhaan, Jan Eeckhout, Ester Faia, Claudio Michelacci, Kris Nimark, Evi Pappa, Giorgio Prim-iceri, Aysegül Sahin, Carlos Thomas, Jaume Ventura, Joachim Voth, Eran Yashiv and participants inthe CREI Macro Faculty Lunch, NBER Summer Institute, EES research network workshop in Kiel,CEPR-RIETI Workshop on �Labor Markets and the Macroeconomy� in Tokyo, University of Amster-dam, Malaga Workshop on �Labor Market Frictions�, European Central Bank, Norges Bank, CPB,ECB-CEPR Workshop on �European Labor Market Adjustment�, Sabanci University, ESSIM 2010 andSED 2010. Davide Debortoli provided outstanding research assistance. We gratefully acknowledge �-nancial support from the European Research Council, the Spanish Ministry of Science and Innovation(grants SEJ2006-02235, ECO2008-01665, CSD2006-00016 and Juan de la Cierva); the Generalitat deCatalunya (DIUE grant 2009SGR1157); and the Barcelona GSE Research Network.
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1 Introduction
The nature of business cycle �uctuations changes over time. There is a host of evi-
dence for changes in the dynamics of postwar US macroeconomic time series (Blanchard
and Watson (1986), McConell and Pérez-Quirós (2000), Stock and Watson (2002), Hall
(2007), Galí; and Gambetti (2009)). The present paper documents and discusses three
aspects of these changes:
� The correlation of labor productivity with output or labor input has declined, by
some measures dramatically so.1
� The volatility of labor input measures has increased (relative to that of output).2
� The volatility of real wage measures has increased, both in relative and absolute
terms.3
All three of the above observations point towards a change in labor market dynamics.
While each may be of independent interest and have potentially useful implications for
our understanding of macro �uctuations, our goal in the present paper is to explore their
possible connection. In particular, we seek to investigate the hypothesis that all three
changes may be driven by an increase in labor market �exibility, allowing �rms to adjust
their labor force more easily in response to various kinds of shocks. In order to illustrate
the mechanism behind this explanation, we develop a stylized model of �uctuations with
labor market frictions, and investigate how its predictions vary with the parameter that
indexes the importance of such frictions.
The main intuition behind that mechanism is easy to describe. The idea goes back to
a literature, starting with Oi (1962) and Solow (1964), which attributes the procyclicality
of productivity to variations in e¤ort, resulting in seemingly increasing returns to labor.4
Suppose that �rms have two margins for adjusting their e¤ective labor input: (observed)
employment and (unobserved) e¤ort, which we respectively denote (in logs) by nt and
1As far as we know, Stiroh (2009) was the �rst to provide evidence of a decline in the laborproductivity-hours correlation. Gordon (2010), Barnichon (2008), Galí; and Gambetti (2009), andNucci and Riggi (2009), using di¤erent approaches, independently investigated the potential sources ofthat decline.
2To the best of our knowledge, Galí; and Gambetti (2009) were the �rst to uncover that �nding, butdid not provide the kind of detailed statistical analysis found below. Independently, Hall (2007) o¤eredsome evidence on the size of the decline in employment in the most recent recessions that is consistentwith our �nding.
3As far as we know, this �nding was not known previously, although it is reported in independentwork by Gourio (2007) and Champagne and Kurmann (2010).
4Contributions include studies by Fair (1969), Fay and Medo¤ (1985), Hall (1988), Rotemberg andSummers (1990), Bernanke and Parkinson (1991), Shapiro (1993), Burnside, Eichenbaum, and Rebelo(1993), Bils and Cho (1994), Uhlig and Xu (1996), Basu (1996), Basu and Fernald (1997), Basu andKimball (1997), Shea (1999), Gordon (2004), Wen (2004), Arias, Hansen, and Ohanian (2007), andGordon (2010)
2
et.5 Labor input (employment and e¤ort) are transformed into output according to a
standard production function,
yt = (1� �)(nt + et) + at
where at is log total factor productivity and � is a parameter measuring diminishing
returns to labor.
Measured labor productivity, or output per person, is given by
yt � nt = ��nt + (1� �) et + at
Labor market frictions make it costly to adjust employment nt. E¤ort et provides an
alternative margin of adjustment of labor input and is not subject to those frictions (or
to a lesser degree). Thus, the larger the frictions, the less employment �uctuates and
the more volatile �uctuations in e¤ort. As a result, a reduction in frictions decreases
the volatility of e¤ort and therefore increases the relative volatility of employment with
respect to output. The increased volatility of nt also makes labor productivity less
procyclical, and, in the presence of shocks other than shifts in technology, may even
make productivity countercyclical, consistent with the evidence reported below.
In addition, as emphasized by Hall (2005), the presence of labor market frictions
generates a non-degenerate bargaining set for the wage, i.e. a wedge between �rms�
and workers� reservation wages. Any wage within that bargaining set is consistent with
labor market equilibrium. That feature makes room for wage rigidities. We model
wages as rigid within the bargaining set, adjusting only when approaching its bounds.
In our model, a reduction in labor market frictions narrows the bargaining set and
therefore endogenously makes wages more sensitive to shocks, increasing the volatility of
�uctuations in wages. If the rigidity is extended to the wages of newly hired workers, then
the increased �exibility of wages may dampen the volatility of output and employment in
response to shocks.6 That feature may help explain the observed decline in the volatility
of those two variables in the recent US experience.7 Other authors have also argued
that the Great Moderation may have been driven at least in part by increased wage
�exibility (Gourio (2007), Champagne and Kurmann (2010), Nucci and Riggi (2009)).
However, this paper is the �rst to show that such an increase in wage �exibility can
arise endogenously from a decrease in labor market frictions.8
5To simplify the argument, we assume hours per worker are constant, consistent with the observationthat in the U.S. data most adjustments in total hours worked take place along the extensive margin.
6This is clearly true for technology shocks. As argued in Blanchard and Galí (2008), increased wage�exibility may also dampen the sensitivity of GDP and in�ation to oil price shocks.
7A more �exible labor market does of course not make the economy immune to very large shocks likethe recent �nancial crisis. Under our hypothesis, if the labor market were as rigid as it was in the early80s, the current recession might have been even substantially more severe.
8Uren (2008) develops a model, in which a reduction in search frictions decreases output volatility.However, the mechanism in that paper (increased assortative matching) is completely di¤erent.
3
The remainder of the paper is organized as follows. Section 2 documents the changes
in the patterns of �uctuations in labor productivity, employment and wages. Section
3 develops the basic model. Section 4 describes the outcome of simulations of a cali-
brated version of the model, and discusses its consistency with the evidence. Section 5
concludes.
2 Changes in Labor Market Dynamics
We document three stylized facts regarding postwar changes in US economic �uctuations
that motivate our investigation. All three facts are about changes in the dynamics of
the labor market and pertain to the cyclical behavior of labor productivity, labor input
and wages. We use quarterly time series over the period 1948:1-2007:4 drawn from
di¤erent sources (see below for a detailed description). To illustrate the changes in the
di¤erent statistics considered, we split the sample period into two subperiods, pre-84
(1948:1-1983:4) and post-84 (1984:1-2007:4). The choice of the break date is motivated
by existing evidence on the timing of the Great Moderation, the sharp drop in output
volatility around 1984 (McConell and Pérez-Quirós (2000)).
Our evidence makes use of alternative measures of output and labor input. In all
cases labor productivity is constructed as the ratio between the corresponding output
and labor input measures. Most of the evidence uses output and hours in the private
sector from the BLS Labor Productivity and Cost (LPC) program. We also use GDP as
an economy-wide measure of output, with the corresponding labor input measures being
total hours or employment. The time series for economy-wide hours is an unpublished
series constructed by the BLS and used in Francis and Ramey (2008). The employment
series is the usual one from the Current Employment Statistics (CES) establishment
survey. In all cases we normalize the output and labor input measures by the size of the
civilian noninstitutional population (16 years and older).
We apply two alternative transformations on the logarithms of all variables in order
to render the original time series stationary. Our preferred transformation uses the
bandpass (BP) �lter to remove �uctuations with periodicities below 6 and above 32
quarters, as in Stock and Watson (1999). We also apply the fourth-di¤erence (4D)
operator, which is the transformation favored by Stock and Watson (2002) in their
analysis of changes in output volatility.9
2.1 The Vanishing Procyclicality of Labor Productivity
Figure 1 shows the �uctuations at business cycle frequencies in labor productivity in
the US over the postwar period. It is clear from the graph that in the earlier part
9We also showed the results are robust to detrending with the Hodrick-Prescott (HP) �lter andto using shorter sample periods, centered around the break date 1984. These estimates, which aresuppressed for brevity, are available upon request.
4
of the sample, productivity was signi�cantly below trend in each recession. However,
in the post-84 data, this is no longer the case. When we calculate the correlation of
productivity with output or employment, as in Figure 2, it is clear that there is a sharp
drop in the cyclicality of productivity around 1984. The correlation of productivity
with output, which used to be strongly positive, fell to a level close to zero, while the
correlation of productivity with employment, which was zero or slightly positive in the
earlier period of the sample, became negative.
These �ndings are formalized in Table 1, which reports the contemporaneous corre-
lation between labor productivity and output and employment, for alternative transfor-
mations and time periods. In each case, we report the estimated correlation for the pre
and post-84 subsamples, as well as the di¤erence between those estimates. The standard
errors, reported in brackets, are computed using the delta method.10 We now turn to a
short discussion of the results in this Table.
2.1.1 Correlation with Output
Independently of the detrending procedure, the correlation of output per hour with out-
put in the pre-84 period is high and signi�cantly positive, with a point estimate around
0:60. In other words, from the vantage point of the early 80s �the period when the sem-
inal contributions to RBC theory were written� the procyclicality of labor productivity
was a well established empirical fact, which lent support to business cycle theories that
assigned a central role to technology shocks as a source of �uctuations.
In the post-84 period, however, that pattern changed considerably. The estimates
of the productivity-output correlation dropped to a value close to (and not signi�cantly
di¤erent from) zero. The di¤erence with the corresponding pre-84 estimates is highly
signi�cant. Thus, on the basis of those estimates labor productivity has become an
acyclical variable (with respect to output) over the past two decades.
When we use an employment-based measure of labor productivity, output per worker,
the estimated correlations also drop substantially but remain signi�cantly greater than
zero in the post-84 period. This should not be surprising given that hours per worker are
highly procyclical in both subperiods and that their volatility relative to employment-
based labor productivity has increased considerably.11
10We use least squares (GMM) to estimate the second moments (variances and and covariances) ofeach pair of variables, as well as the (asymptotic) variance-covariance matrix of this estimator. Then,we calculate the standard errors for the standard deviations, the relative standard deviations and thecorrelation coe¢cient using the delta method.11Letting n and h denote employment and total hours respectively, a straightforward algebraic ma-
nipulation yields the identity:
�(y � n; y) =�y�h�y�n
�(y � h; y) +�h�n�y�n
�(h� n; y)
Thus, even in the case of acyclical hours-based labor productivity, i.e. �(y � h; y) ' 0, we would expect�(y � n; y) to remain positive if hours per worker are procyclical, i.e. �(h� n; y) > 0.
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2.1.2 Correlation with Labor Input
The right-hand side panels in Table 1 display several estimates of the correlation between
labor productivity and labor input. The estimates for the pre-84 period are low, and in
the case of output per hour, insigni�cantly di¤erent from zero. Thus, labor productivity
was largely acyclical with respect to labor input in that subperiod. This near-zero
correlation is consistent with the evidence reported in the early RBC literature, using
data up to the mid 80s.12
As was the case when using output as the cyclical indicator, the estimated correla-
tions between labor productivity and employment decline dramatically in the post-84
period. In fact these correlations become signi�cantly negative for output per hour, with
a point estimate ranging from �0:40 to �0:54, depending on the �lter. The change with
respect to the pre-84 period is highly signi�cant. In other words, labor productivity in
the past two decades has become strongly countercyclical with respect to labor input.
2.2 The Rising Relative Volatility of Labor Input
The left-hand panel of Table 2 displays the standard deviation of several measures
of labor input in the pre and post-84 periods, as well as the ratio between the two.
The variables considered include employment in the private sector, hours in the private
sector (employment times hours per worker) and economy-wide hours. The decline in
the volatility of hours since the mid 80s, like that of other major macro variables, is seen
to be large and highly signi�cant, with the standard deviation falling between 35% and
49% and always signi�cantly so.
A more interesting piece of evidence is the change in the relative volatility of labor
input, measured as the ratio of the standard deviation of labor input to the standard
deviation of output. These estimates are presented in the right-hand panel of Table
2. Without exception, all labor input measures have experienced an increase in their
relative volatility in the post versus pre-84 period. In other words, the decline in the
variability of labor input has been less pronounced than that of output. The increase in
the relative volatility of hours worked ranges from 30% to 48% in the private sector and
from 7% to 30% in the total economy. The corresponding increase for employment is
slightly smaller, ranging from 23% to 43% in the private sector, but is still statistically
signi�cant.
The previous evidence points to a rise in the elasticity of labor input with respect to
output. Put di¤erently, �rms appear to have relied increasingly on labor input adjust-
ments in order to meet their changes in output.
12Christiano and Eichenbaum (1992) used data up to 1983:4 (which coincides with the cut-o¤ date forour �rst subperiod), but starting in 1955:4. Their estimates of the correlation between labor productivityand hours were �0:20 when using household data and 0:16 using establishment data.
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2.3 The Rising Volatility of Wages
Next we turn our attention to the volatility of (real) wages, both in absolute and relative
terms. We consider four di¤erent wage measures. The �rst three are constructed as real
compensation per hour. The �rst di¤erence is in the measure of compensation, which is
measured either from the national income and product accounts (NIPA) or as earnings
from the CES establishment survey. The second di¤erence is in the measure of hours,
which refers to the private sector or to the total economy.13 The fourth measure is
usual hourly earnings (or usual weekly earnings divided by usual weekly hours) from
the Current Population Statistics (CPS). For all measures, compensation or earnings
are de�ated using the compensation de�ator from the LPC, but the results are robust
to de�ating with the consumer price index (CPI-U) as we show below.
2.3.1 Average Wages
The left-hand panel of Table 3 displays the standard deviation for each wage measure for
di¤erent detrending procedures. Our statistics uncover a surprising �nding: despite the
general decline in macro volatility associated with the Great Moderation, the volatility
of several wage measures has increased in absolute terms. The estimated increase the
standard deviation is fairly large, between 10 and 42% and mostly signi�cant for the
bandpass �ltered NIPA-based wage measures. Using fourth di¤erences, the increase is
much smaller and no longer signi�cant and by some periods there seems to have been a
(small) decrease in wage volatility.
Using earnings per hour from the CES, however, there seems to be a large and highly
signi�cant reduction in wage volatility.14 One di¤erence between the two measures is
that the NIPA compensation measure includes non-wage payments and, in particular,
employee stock options. Mehran and Tracy (2001) have argued that since these options
are recorded when they realize rather than when they are handed out to employees, the
NIPA measure gives a misleading picture of the evolution and volatility of compensation
in the 90s. However, using the CPS measure of usual hourly earnings, presented in
Table 4, which includes non-wage compensation but not stock options, we again observe
a fairly large increase in the volatility of wages.15 Given the short time series available
13Our baseline measure uses compensation from the NIPA and hours in the private sector and corre-sponds to compensation per hour from the BLS labor productivity and cost program (LPC).14This �nding is not driven by the fact that the CES earnings measure is only available after 1965. If
we restrict the sample for the NIPA based wage measures to the 1965-2004, the volatility statistics forthese measures look very similar those in the table.15We use data from the CPS outgoing rotation groups. Since these data are available only from
1979 onwards, we compare the volatility over the 1980-1984 period (allowing for fourth di¤erences) withthat of the 1985-2005 period. For comparison, the �rst panel of Table 4 presents the volatility of ourbaseline measure for compensation per hour for this period. The second panel presents comparablestatistics from the CPS series. Because the CPS wage series is based on a fairly small cross-section ofworkers, there is substantial measurement error in these series. Therefore, the standard deviations ofthe fourth di¤erenced data are biased upward, see Haefke, Sonntag, and van Rens (2008) for details.
7
for these data, it is remarkable that the increase in volatility is (borderline) signi�cant
for the bandpass �ltered series. Using fourth di¤erences, wage volatility seems roughly
constant.
Our �nding that wage volatility increased or at least did not decrease around the
time of the Great Moderation, although with a caveat, is consistent with the results in
Champagne and Kurmann (2010), who also use the CPS to show that the increase in
wage volatility is not driven by compositional changes in the labor force. To the best of
our knowledge, this result was not previously known.16
An immediate implication of the previous �nding, and the one that we want to
emphasize here, is the possibly very large increase in the relative volatility of wages
with respect to to output or labor input, as shown in the right-hand panels of Tables
3 and 4. The relative volatility of wages with respect to output more than doubled for
the NIPA-based measures and for the CPS wage. We interpret this evidence as being
consistent with a decline in the signi�cance of real wage rigidities around 1984.17
2.3.2 Wages of Newly Hired Workers
On a frictional labor market, the average wage is not allocative, since the frictions drive
a wedge between the reservation wages of �rms and workers. Therefore, in order to
assess the implications of increased wage �exibility for other labor market variables, we
also consider the volatility of the wage of newly hired workers as suggested by Haefke,
Sonntag, and van Rens (2008) and Pissarides (2009).
Table 4 presents volatility statistics for the wage of new hires, constructed from the
CPS as in Haefke, Sonntag, and van Rens (2008).18 The �rst thing to notice is that the
wage of newly hired workers is much more volatile than the average wage in the entire
labor force. This is consistent with the results in Haefke, Sonntag, and van Rens (2008),
who argue that, in the post-84 period, wages of newly hired workers are perfectly �exible,
in the sense that they respond one-to-one to changes in labor productivity. Here, we
focus on the change in the volatility of wages over time.
The absolute volatility of the wage of newly hired workers, unlike the average wage,
decreased substantially and signi�cantly between the pre and post-84 periods. As a
result, the increase in the relative volatility with respect to output is much smaller for
There is no reason however, why the ratio of the standard deviations before and after 1984 would bebiased. In addition, the bandpass �ltered data, which do not include the high frequencies induced bythe measurement error, are not subject to this bias.16Stock and Watson (2002) uncover breaks in the volatility of a long list of macro variables, but they
do not provide evidence for any wage measure.17Blanchard and Galí (2008) argue that a reduction in the rigidity of real wages is needed in order to
account for the simultaneous decline in in�ation and output volatility, in the face of oil price shocks ofa similar magnitude.18But unlike in that paper, we do not correct �uctuations in the CPS wage series for changes in
the composition of the labor force by demographic characteristics, education level and experience forcomparability with the other wage measures. Doing so however, makes very little di¤erence for theconclusions presented here.
8
new hires, ranging between 3% and 69%, depending on whether we use the mean or
median wage and on the �lter used. Although the increase in the relative volatility of
the wage of newly hired workers is much less pronounced, there is some evidence that
wages �uctuated more between recessions and booms also for this group of workers.
This �nding is consistent with the evidence presented in Haefke, Sonntag, and van Rens
(2008, section 3.4) and points towards a decrease in wage rigidity that may be important
for employment �uctuations.
2.4 Conclusions
Summarizing, we showed that labor productivity became less procyclical or acyclical
with respect to output, and countercyclical with respect to employment. In addition,
the relative volatility of both employment and wages increased. For completeness, we
also report that the relative volatility of labor productivity increased, and the correla-
tion between employment and output decreased slightly, see Table 5.19 These changes
coincided with the reduction in volatility in output and most other macroeconomic
aggregates, the so called Great Moderation. This evidence is suggestive of structural
change in the labor market.
In macroeconomic models with a perfectly competitive labor market and a standard
production function, wages are proportional to labor productivity.20 Our evidence makes
clear that the extent to which this is true in the data depends very much on the period
one looks at. From Tables 3 and 5, we see that the relative standard deviation of wages
with respect to labor productivity was about 0:3� 0:7 in the the pre-84 period. In the
post-84 period, this relative standard deviation more than doubled to about 1:2 � 1:3.
This is consistent with the evidence in Haefke, Sonntag, and van Rens (2008, section
3.4), who show that the elasticity of wages with respect to productivity in 1984 increases
from 0:2 to 0:4 for all workers, and from 0:3 to 0:8 for newly hired workers. This evidence
suggests that labor market frictions are crucial to understand the observed changes in
labor market dynamics.
In the remainder of this paper, we will show that the vanishing procyclicality of labor
productivity and the increasing relative volatility of employment, can be explained by
a reduction in labor market frictions, if production requires an unobservable input, e.g.
e¤ort. If wages are set competitively, then a reduction in labor market frictions, which
makes employment faster to adjust, should make wages relatively less volatile. Although
the relative standard deviation of wages decreased in some datasets, we argued above
that the majority of the evidence seems to point towards a large increase in the relative
19These observations are completely determined by the statistics already reported and do not containindependent information We emphasize the statistics that we considered easiest to interpret.20With competitive labor markets, the wage equals the marginal product of labor, and with a Cobb-
Douglas production function, the marginal and average product of labor are proportional to eachother.This well-known argument, which does not rely on business cycles being driven by productivity shocks,was used recently by Rogerson and Shimer (2010, section 1.3.2).
9
volatility of wages. Therefore, we show below how the model can be extended with
endogenous wage rigidities in order to generate this result.
3 A Model of Fluctuations with Labor Market Frictions
and Endogenous E¤ort
Having documented in some detail the changing patterns of labor productivity, labor in-
put, and wages, we turn to possible explanations. More speci�cally, and as anticipated
in the introduction, we explore the hypothesis that all three observed changes docu-
mented above may have, at least partly, been caused by the same institutional change:
increasing �exibility of the labor market.
To formalize this explanation, we develop a model of �uctuations with labor market
frictions, modelled as adjustment costs in employment (hiring costs). The crucial element
in this model is an endogenous e¤ort choice, which provides an intensive margin for
labor adjustment that is not subject to the adjustment costs. Since the purpose of the
model is to illustrate the main mechanisms at work, we keep the model as simple as
possible in dimensions that are likely to be orthogonal to the factors emphasized by
our analysis. Thus, we abstract from endogenous capital accumulation, trade in goods
and assets with the rest of the world, and imperfections in the goods and �nancial
markets. We also ignore any kind of monetary frictions, even though we recognize that
these, in conjunction with changes in the conduct of monetary policy in the Volcker-
Greenspan years, may have played an important role in accounting for the decline in
macro volatility.21
3.1 Households
Households are in�nitely-lived and consist of a continuum of identical members repre-
sented by the unit interval. The household is the relevant decision unit for choices about
consumption and labor supply. Each household member�s utility function is additively
separable in consumption and leisure, and the household assigns equal consumption Ct
to all members in order to share consumption risk within the household. Thus, the
household�s objective function is given by,
E0
1X
t=0
�t
"
ZtC1��t
1� �� Lt
#
(1)
where � 2 (0; 1) is the discount factor, � 2 [0; 1] is the inverse of the intertemporal
elasticity of substitution, > 0 can be interpreted as a �xed cost of working and Zt
is a preference shock. The second term in the period utility function is disutility from
21See, e.g. Clarida, Galí, and Gertler (2000) for a discussion of the possible role of monetary policyin the Great Moderation.
10
e¤ective labor supply Lt, which depends on the fraction Nt of household members that
are employed, as well as on the amount of e¤ort Eit exerted by each employed household
member i. Formally,
Lt =
Z Nt
0
1 + �E1+�it
1 + �di =
1 + �E1+�t
1 + �Nt (2)
where the second equality imposes the equilibrium condition that all working household
members exert the same level of e¤ort, Eit = Et for all i. The parameter � � 0 measures
the importance of e¤ort for the disutility of working, and the elasticity parameter � � 0
determines the degree of increasing marginal disutility from exerting e¤ort. For simplic-
ity we assume a constant workweek, thus restricting the intensive margin of labor input
adjustment to changes in e¤ort.
The household maximizes its objective function above subject to the sequence of
budget constraints,
Ct =
Z Nt
0Witdi+�t (3)
where �t represents �rms� pro�ts, which are paid out to households in the form of
lump-sum dividents, and Wit are wages accruing to employed household member i. The
household takes into account the e¤ect of its decisions on the level of e¤ort exerted by
its members.
3.2 Firms
Firms produce a homogenous consumption good using a production technology that
uses labor and e¤ort as inputs,
Yt = At
�Z Nt
0E itdi
�1��
= At
�
E t Nt
�1��(4)
where Yt is output, Eit is e¤ort exerted by worker i, � 2 (0; 1) is a parameter that
measures diminishing returns to total labor input in production, 2 [0; 1] measures
additional diminishing returns to e¤ort, and At is a technology shock common to all
�rms. Since all �rms are identical, we normalize the number of �rms to the unit interval,
so that Yt and Nt denote output and employment of each �rm as well as aggregate output
and employment in the economy. The second equality imposes the equilibrium condition
that all workers in a �rm exert the same level of e¤ort, Eit = Et for all i.
Firms choose how many workers to hire Ht in order to maximize the expected dis-
counted value of pro�ts,
E0
1X
t=0
Q0;t [Yt �WtNt � g (Ht)] (5)
11
subject to a law of motion for employment implied by the labor market frictions,
Nt = (1� �)Nt�1 +Ht (6)
where the function g (:), with g0 > 0 and g00 > 0, represents the costs (in terms of output)
of hiring new workers and Q0;t is the stochastic discount factor for future pro�ts. The
stochastic discount factor is de�ned recursively as Q0;t � Q0;1Q1;2:::Qt�1;t, where
Qt;t+1 � �Zt+1Zt
�
CtCt+1
��
(7)
measures the marginal rate of substitution between two subsequent periods. Like the
household, the �rm takes into account the e¤ect of its decisions on the level of e¤ort
exerted by its workers.
3.3 E¤ort Choice and Job Creation
The household and the �rm jointly decide the wage and the level of e¤ort that the worker
will put into the job. In equilibrium, the e¤ort level of all workers is set e¢ciently,
maximizing the total surplus generated by each match.22 This e¢cient e¤ort level, in
each period and for each worker, equates the cost of exerting more e¤ort, higher disutility
to the household, to the bene�t, higher production and therefore pro�ts for the �rm.
Consider a worker i, who is a member of household h and is employed in �rm j. The
marginal disutility to the household from that worker exerting more e¤ort, expressed in
terms of consumption, is obtained from equation (2) for total e¤ective labor supply and
equals:
C�htZt
@Lht@Eit
=(1 + �) �
1 + �
C�htE�it
Ztdi (8)
The marginal product of that additional e¤ort to the �rm is found from production
function (4):
@Yjt@Eit
= (1� �) At
�Z Njt
0E vtdv
���
E�(1� )it di (9)
In equilibrium, the marginal disutility from e¤ort must equal its marginal product for
all workers i. Also, because all �rms and all households are identical, it must be that
Cht = Ct and Njt = Nt in equilibrium. Therefore, it follows that all workers exert the
same level of e¤ort in equilibrium, Eit = Et for all i. Imposing this property, we obtain
the following equilibrium condition for e¤ort,
Et =
�
(1� �) (1 + �)
(1 + �) �
Zt C�t
AtN��t
� 11+��(1��)
(10)
22Suppose not. Then, household and �rm could agree on a di¤erent e¤ort level that increases totalmatch surplus, and a modi�ed surplus sharing rule (wage) that would make both parties better o¤.
12
or, using production function (4) to simplify:
E1+�t =
1 + �
1 + �
�
Zt C�t
(1� �)YtNt
(11)
When considering whether to hire a worker, �rms take into account the impact of
the resulting increase in employment on the e¤ort level exerted by their workers. Thus,
the marginal product of a new hire is given by,23
dYjtdNjt
=@Yjt@Njt
+@Yjt@Ejt
@Ejt@Njt
= (1�F )(1� �)Yt
Nt(12)
where F =�
1+��(1��) measures the additional (negative) e¤ect from a new hire on
output that comes from the endogenous response of the e¤ort level in the �rm.
Maximizing the expected net present value of pro�ts (5), where output is given by
production function (4) and the stochastic discount factor by (7), subject to the law
of motion for employment implied by the matching technology (6) and the equilibrium
condition for e¤ort (11), gives rise to the following �rst order condition,
g0 (Ht) = SFt (13)
where SFt is the marginal value to the �rm of having an additional worker in period t,
which is given by,
SFt = (1�F )(1� �)Yt
Nt�Wt + (1� �)Et
�
Qt;t+1SFt+1
�
(14)
= Et
1X
s=0
(1� �)sQt;t+s
�
(1�F )(1� �)Yt+s
Nt+s�Wt+s
�
(15)
where the second equality follows from iterating forward (and de�ning Qt;t = 1). This
is a job creation equation, which states that the marginal costs of hiring a new worker
g0 (Ht), must equal the expected net present value of marginal pro�ts (additional output
minus the wage) of the �lled job, SFt .
3.4 The Bargaining Set
Because of labor market frictions, employment relationships generate a strictly positive
surplus. The reason is that if �rm and worker cannot agree to continue their relationship,
23With a slight abuse of notation, Ejt denotes the e¤ort level exerted by all workers (from di¤erenthouseholds) in a particular �rm j. Firm j considers employing Njt workers, given that all other �rmsemploy the equilibrium number of workers Nt. Because there are in�nitely many �rms, �rm j�s decisionto employ Njt 66= Nt workers does not a¤ect the fraction of household h�s members that are employed, sothat by the assumption of perfect risk-sharing within the household, the consumption of workers in �rmj, Cht = Ct, is not a¤ected. Therefore, the relation between e¤ort and employment that the �rm facesif all other �rms (and all households) play equilibrium strategies, is given by equation (10), keeping Ct�xed. See appendix A for details on the derivation of equation (12).
13
then the �rm has to pay the hiring costs again in order to �nd another worker to match
with. Firms and households bargain over the wage as a way to share the match surplus.
These negotations are limited only by the outside option of each party. The lower bound
of the bargaining set is given by the reservation wage of the household, the wage o¤er at
which the household is indi¤erent between accepting the o¤er and looking for another
job. Similarly, the upper bound of the bargaining set is the reservation wage of the �rm,
the wage o¤er that makes the �rm indi¤erent between accepting the o¤er and hiring
a di¤erent worker. Within these bounds, any wage is consistent with equilibrium, see
Hall (2005). Clearly, the bounds of the bargaining set are endogenous variables, which
we now derive before introducing an equilibrium selection rule for the wage within the
bargaining set.
The part of the match surplus that accrues to the �rm SFt , as a function of the wage,
is given by equation (14). In order to derive a similar expression for the household�s part
of the surplus SHt , we must �rst calculate the marginal disutility to the household of
having one additional employed member, taking into account the endogenous response
of e¤ort. This marginal disutility of employment, expressed in terms of consumption, is
given by,24
C�tZt
dLhtdNht
=1
1 + �
C�tZt
�
1 + �(1 + �)H
E1+�t
�
=1
1 + �
C�tZt
+H(1� �)Yt
Nt(16)
where the second equality follows from substituting equation (11), and where H = 1+�
(1��)(1+�)� 1+�� captures the e¤ect on utility of one more employed member in the
household through the endogenous response of e¤ort. Using this expression, we can
take a derivative of the household�s objective function (1) with respect to Nt and divide
by the marginal utility of consumption, to obtain the following expression for SHt .
SHt =Wt �1
1 + �
C�tZt
�H(1� �)Yt
Nt+ (1� �)Et
�
Qt;t+1SHt+1
�
(17)
The value to the household of having one more employed worker, equals the wage minus
the disutility expressed in terms of consumption, plus the expected value of still having
that worker next period, which is discounted by the probability that the worker is still
employed next period.
The upper bound of the bargaining set WUBt is the highest wage such that SFt � 0,
whereas the lower bound WLBt is the lowest wage such that SHt � 0. Using equations
(14) and (17), we get SFt = WUBt �Wt and S
Ht = Wt �WLB
t . Substituting back into
equations (13), (14) and (17), we can explicitly write the equilibrium of the model in
terms of the wage and the bounds of the bargaining set.
g0 (Ht) =WUBt �Wt (18)
24The derivation of this expression is similar to that of equation (12), see appendix A for details.
14
WUBt = (1�F )
(1� �)YtNt
+ (1� �)Et�
Qt;t+1�
WUBt+1 �Wt+1
��
(19)
WLBt =
1
1 + �
C�tZt
+H(1� �)Yt
Nt+ (1� �)Et
�
Qt;t+1�
WLBt+1 �Wt+1
��
(20)
Everything else equal, the more rigid is the wage in response to technology or preference
shocks that shift the bounds of the bargaining set, the more volatile is hiring Ht and
therefore employment Nt in response to those shocks, see equation (18). We now turn
to various possibilities for how wages are determined within the bargaining set.
3.5 Wage Determination
One possible criterion for wage determination that we can interpret as �exible wages
in a model with a frictional labor market, is period-by-period Nash bargaining. Nash
bargaining assumes that the wage is set such that the total surplus from the match is
split in equal proportions between household and �rm. It is straightforward to see that
in our framework, this assumption implies that the wage is the average of the lower and
upper bounds of the bargaining set, SHt = 12
�
SHt + SFt
�
= 12
�
WUBt �WLB
t
�
. Then,
W �
t =12
�
WUBt +WLB
t
�
(21)
where W �
t denotes the Nash bargained wage.
Shimer (2005) and Hall (2005), among others, have argued that period-by-period
Nash bargaining generates too volatile a wage in equilibrium, relative to what is observed
in the data. As discussed below, in our model period-by-period Nash bargaining leads to
�uctuations in the (log) wage of the same amplitude as labor productivity, and perfectly
correlated with the latter. This is at odds with the data, where wages are about half
as volatile as labor productivity in the pre-84 period, with the correlation between the
two variables much smaller than one. Both the relative volatility of wages and their
correlation with labor productivity increases signi�cantly in the post-84 period. This
motivates the introduction of a wage setting mechanism that departs from period-by-
period Nash bargaining.25 We use the following wage determination process,
Wt = Rt�1Wt�1 + (1�Rt�1)W�
t (22)
where Rt measures the degree of wage rigidity, which is endogenous.
Only wages within the bargaining set are renegotiation-proof and therefore consistent
with equilibrium, see Barro (1977). The maximum degree of wage stickiness that respects
this condition, is for the wage to remain �xed inside the bargaining set, but to be adjusted
25This is consistent with the recent trend in the literature, see e.g. Hall (2005), Blanchard and Galí(2007), Hall and Milgrom (2008), Christo¤el and Kuester (2008), Gertler and Trigari (2009), and Shimer(2009).
15
when it hits either bound. This is the wage determination mechanism in Thomas and
Worrall (1988) and Hall (2003). We use a convex version of this wage rule, in order
to be able to solve the model using perturbation methods, and assume the following
reduced-form equation for wage rigidity,
Rt = �R
0
@1�
Wt �W�
t12
�
WUBt �WLB
t
�
!2�1
A (23)
where � 2 N+0 . This wage rule captures the idea that the wage is more likely to adjust
when it is closer to the bounds of the bargaining set. The parameter � captures the
degree of non-linearity in this relation. For � = 0, Rt = 0 and Wt = W �
t , i.e. wages
are �exible. For � 2 N+, the degree of wage rigidity is endogenous, with wages being
perfectly �exible at the upper or lower bound of the bargaining set and most rigid at
the Nash-bargained wage W �
t . As � becomes larger, wages are rigid in a larger part of
the bargaining set. The limiting case for � ! 1 and �R = 1 captures the case where
the wage is �xed within the bargaining set but adjusts when it has to in order to avoid
ine¢cient match destruction as in Hall (2003). We consider a �exible wage regime with
� = 0 and regime with endogenous wage rigidity, � 2 N+ and �R = 1.
The crucial insight for our purposes is that with this type of wage rule, the degree
of wage rigidity depends endogenously on the size of the frictions. If frictions decrease,
the width of the bargaining set decreases as well, so that there is less room for wage
rigidity. Notice also that this type of wage rigidity can never lead to ine¢cient match
destruction.
3.6 Equilibrium
We conclude the description of the model by listing the conditions that characterize the
equilibrium. Vacancy posting decisions by �rms are summarized by the job creation
equation (18).
g0 (Ht) =WUBt �Wt (24)
The equilibrium level of e¤ort is determined by e¢ciency condition (11),
E1+�t =
1 + �
1 + �
�
Zt C�t
(1� �)YtNt
(25)
and wage negotations are described by equations (22) and (23) for the equilibrium
selection rule, and stochastic di¤erence equations for the upper and lower bounds of the
bargaining set (19) and (20).
Wt = RtWt�1 + (1�Rt)12
�
WUBt +WLB
t
�
(26)
16
Rt = �R
0
@1�
Wt �12
�
WUBt +WLB
t
�
12
�
WUBt �WLB
t
�
!2�1
A (27)
WUBt = (1�F )
(1� �)YtNt
+ (1� �)Et�
Qt;t+1�
WUBt+1 �Wt+1
��
(28)
WLBt =
1
1 + �
C�tZt
+H(1� �)Yt
Nt+ (1� �)Et
�
Qt;t+1�
WLBt+1 �Wt+1
��
(29)
Employment evolves according to its law of motion (6).
Nt = (1� �)Nt�1 +Ht (30)
Finally, goods market clearing requires that consumption equals output minus hiring
costs.
Ct = Yt � g (Ht) (31)
Output is de�ned as in production function (4), and the stochastic discount factor
as the marginal rate of intertemporal substitution (7).
Yt = At
�
E t Nt
�1��(32)
Qt;t+1 = �Zt+1Zt
�
CtCt+1
��
(33)
and the parameters F =�
1+��(1��) and H = 1+�
(1��)(1+�)� 1+�� are functions of the
structural parameters. In total, we have 8 equations in the endogenous variables Ht,
Et, Wt, Rt, WUBt , WLB
t , Nt and Ct, or 10 equations including the de�nitions for Yt and
Qt;t+1.
Without an endogenous e¤ort choice ( = 0 so that e¤ort is not useful in production,
F = H = 0, and Et = 0 for all t in equilibrium), and with �exible wages ( �R = 0 so
that Rt = 0 for all t), the model reduces to a standard RBC model with labor market
frictions. However, unlike in the standard model, �uctuations in our model are driven
by technology shocks as well as non-technology shocks or preference shocks. The two
driving forces of �uctuations, log total factor productivity at � logAt and log preferences
over consumption zt � logZt follow stationary AR(1) processes,
at = �aat�1 + "at (34)
zt = �zzt�1 + "zt (35)
where "at and "zt are independent white noise processes with variances given by �
2a and
�2z respectively.
We now proceed to use this model to analyze the possible role of labor market
17
frictions in generating the observed changes in the cyclical patterns of output, labor
input, productivity, and wages. For this analysis, all three above-mentioned elements
are important, i.e. multiple shocks, endogenous e¤ort and endogenous wage rigidity.
4 The Increasing Flexibility of the Labor Market
This section provides an analysis of our model economy�s equilibrium under alternative
assumptions regarding the size of labor market frictions and wage determination. We
start by looking at a version of the model with a frictionless labor market. This model
provides a useful benchmark that we can solve for in closed form. Then, we introduce
frictions and rely on numerical methods to simulate the model for di¤erent calibrations
of the parameters.
4.1 The Frictionless Case
Consider the limiting case of an economy without labor market frictions, i.e. g (H) = 0
for all H. The �rst thing to note is that in this case the width of the bargaining set
collapses to zero, and the job creation equation (24) and the wage block of the model,
equations (26), (27), (28) and (29), imply
Wt =WUBt =WLB
t = (1�F )(1� �)Yt
Nt=
1
1 + �
C�tZt
+H(1� �)Yt
Nt(36)
for all t. Employment becomes a choice variable, so that its law of motion (30) is dropped
from the system and employment is instead determined by the static condition (36).
Nt = (1� �) (1�F �H)(1 + �)ZtYt
C�t(37)
Substituting into the equilibrium condition for e¤ort (25), we obtain
E1+�t =
1 + �
1
�
1
1�F �H(38)
implying an e¤ort level that is invariant to �uctuations in the model�s driving forces.
Since e¤ort has stronger diminishing returns in production and stronger increasing mar-
ginal disutility than employment, this intensive margin of adjustment is never used if
the extensive margin is not subject to frictions.
Without hiring costs, the aggregate resource constraint (31) reduces to Ct = Yt.
Combining the resource constraint and equations (37) and (38) with the production
function (32), we can derive closed-form expressions for equilibrium employment, output,
wages and labor productivity. Using lower-case letters to denote the natural logarithms
of the original variables, ignoring constant terms and normalizing the variance of the
18
shocks,26 we get:
nt = (1� �) at + zt (39)
yt = at + (1� �) zt (40)
wt = yt � nt = �at � �zt (41)
A useful benchmark is the model with logarithmic utility over consumption (� = 1). In
this case, employment �uctuates in proportion to the preference shifter zt but does not
respond to technology shocks.27
From the previous equations, it is straightforward to calculate the model�s implica-
tions for the second moments of interest. In particular we have
cov (yt � nt; yt) = � var (at)� � (1� �) var (zt) (42)
cov (yt � nt; nt) = � (1� �) var (at)� � var (zt) (43)
In the absence of labor market frictions, labor productivity is unambiguously counter-
cyclical in response to preference shocks. The intuition for this result is that output
responds to preference shocks only through employment, and this response is less than
proportional because of diminishing returns in labor input (� > 0). Since productivity
is unambiguously procyclical in response to technology shocks, the unconditional corre-
lations depend on the relative variances of the shocks and the model parameters. For a
wide range of parameter values, e.g. with logarithmic utility over consumption (� = 1),
productivity is procyclical with respect to output but countercyclical with respect to
employment.
The relative volatility of employment and wages with respect to output are given by
the following expressions:
var (nt)
var (yt)=(1� �)2 var (at) + var (zt)
var (at) + (1� �)2 var (zt)
(44)
var (wt)
var (yt)=
�2 var (at) + �2 var (zt)
var (at) + (1� �)2 var (zt)
(45)
The size of the relative volatility measures above depends again on the relative impor-
tance of the shocks, as well as on the size of �, the parameter determining the degree
of diminishing returns to labor.
26 If the original shocks are ~at and ~zt, then we de�ne at = ~at and zt = ~zt, where =1= [1� (1� �) (1� �)].27This result is an implication of the logarithmic or �balanced growth� preferences over consumption
in combination with the absence of capital or any other intertemporal smoothing technology, and issimilar to the �neutrality result� in Shimer (2009).
19
4.2 In�nite Labor Market Frictions
We can contrast the predictions of the frictionless model above, with the opposite ex-
treme case of in�nitely large labor market frictions, i.e. g (H) = 1 if H > 0. In this
case, no new workers will be hired, so that by the aggregate resource constraint (31)
Ct = Yt, as in the frictionless case. For simplicity, also assume that the separation rate
equals zero, � = 0, so that employment is �xed. In this case, combining the produc-
tion function (32) with the equilibrium condition for e¤ort (25), and taking logarithms,
ignoring constant terms and normalizing the variance of the shocks,28 we get:
et = (1� �) at + zt (46)
yt = yt � nt = (1 + �) at + (1� �) zt (47)
Since employment is �xed, e¤ort is now procyclical in response to both types of shocks, as
all of the adjustment of labor input occurs on the intensive margin. With in�nitely large
labor market frictions, labor productivity is perfectly (positively) correlated with output.
The correlation between productivity and employment, as well as the relative volatility
of employment with respect to output equal zero. Finally, since the bargaining set is
now in�nitely wide, wages may be arbitrarily rigid, depending on the model parameters,
so that the relative volatility of wages is also arbitrarily close to zero.
4.3 Preview of the Results
Comparing these predictions with those of the frictionless model in the previous sub-
section, it is clear that by moving from a completely rigid to a completely �exible labor
market:
1. Labor productivity becomes less procyclical with respect to output.
2. Labor productivity goes from acyclical to countercyclical with respect to employ-
ment, depending on parameter values (a su¢cient condition is logarithmic utility
over consumption).
3. The relative volatility of employment increases.
4. The relative volatility of wages increases.
All four of these predictions are consistent with the data, as we documented in section
2. We are not arguing, of course, that the US labor market went from completely
rigid to completely �exible. Rather, the argument so far is meant to illustrate that if
the reduction in labor market frictions was large enough, it can qualitatively explain
28 In this case, the normalization factor is 1= [1 + �� (1� �) (1� �) ].
20
the patterns we observe in the data. To answer the question whether we can also
quantitatively match those patterns for reasonable parameter values, we now turn to a
numerical analysis of the full model.
4.4 Calibration
We simulate data at quarterly frequency and calibrate accordingly. The calibration is
summarized in Table 6. Many of the model�s parameters can be easily calibrated to
values that are standard in the literature. In this vein, we set the discount factor �
equal to 0:99, assume logarithmic utility over consumption (� = 1), and assume � = 1=3
for the curvature of the production function to match the capital share in GDP.
In the model there is no di¤erence between unemployment and non-participation.
Therefore, we set the marginal utility from leisure to match the employment-population
ratio. Since the amount of labor market frictions a¤ects this ratio as well, we calibrate
to an employment-population ratio of 0:7 in the frictionless model. For the (gross) sep-
aration rate �, we assume 30:6% per quarter, calculated from the monthly worker �ows
in Shimer (2007).29
The calibration of the labor market frictions is crucial for the simulation exercise.
We assume a quadratic cost function, g (H) = 12�H
2, see Yashiv (2010), and consider
a range of values for � such that hiring costs vary from 0 to 3% of output. This range
is based on Silva and Toledo (2009), who estimate the costs of hiring a new worker
to be between 3 and 4:5% of the wage of a newly hired worker, see also Hagedorn
and Manovskii (2008, p.1699). In steady state, the number of newly hired workers
equals �H = � �N per quarter, see (30). Wages, in the frictionless model without e¤ort,
approximately equal the marginal product of labor, �W = (1� �) �Y = �N , see (36). Thus,
hiring costs of 4:5% of the wage translate to total hiring costs of about 1% of output.
For the model�s driving forces, we assume high persistence in both shocks, setting
�a = 0:97 to match the �rst-order autocorrelation in Solow residuals, and �z = 0:97
to make sure that none of the results are driven by di¤erences in persistence. Given
those values, we calibrate �2a and �2z so that the frictionless version of the calibrated
model matches the relative volatility of employment and predicts a standard deviation
of log output of 1%. The �rst target is justi�ed by the observation that in this very
simple model, preference shocks are a stand-in for all sources of misspeci�cation that
result in the unemployment volatility puzzle. The second target is arbitrarily chosen
to emphasize that we consider this model mostly illustrative and not able to generate
29Shimer (2007) �nds a monthly separation rate of sm = 0:034 and a monthly job �nding rate offm = 0:45. Then, the quarterly separation rate, the probability that a worker who is employed at thebeginning of the quarter is no longer employed at the end of the quarter, is given by s = sm (1� fm)
2+(1� sm) sm (1� fm) + (1� sm)
2 sm + s2mfm = 0:0606 and the quarterly job �nding rate equals f =fm (1� sm)
2 + (1� fm) fm (1� sm) + (1� fm)2 fm + f2msm = 0:802. Since workers that are separated
in a given quarter may �nd another job within that quarter, the gross separation rate is given by� = s= (1� f) = 0:306.
21
realistic predictions for the overall level of volatility in the economy.
For the parameters related to e¤ort, we have very little guidance from previous
literature. We normalize � and � such that e¤ort is expressed in utility units and
equals 1 in the frictionless steady state. We treat the curvature of the production
function in e¤ort as a free parameter. Since we are mostly interested to illustrate
the qualitative changes in the business cycle moments that the model can generate, we
pick this parameter such that the model approximately replicates the second moments
in the data. The testable prediction here is not whether the model can quantitatively
match some or most of the second moments, but whether it can qualitatively generate
all observed changes, changing only the labor market frictions.
4.5 Simulation Results
We now simulate the calibrated model in order to calculate the second moments of
interest. We start with the model with �exible wages and show that a reduction in
labor market frictions matches the data on the cyclicality of labor productivity and the
relative volatility of labor input. Then we consider endogenous wage rigidity and show
that the model can also match an increase in the relative volatility of wages, and -if
endogenous wage rigidities are strong enough- can also generate a reduction in output
volatility.
The model with �exible wages is close to log-linear and a �rst-order approximation
captures well the dynamics generated by the model. However, the model with endoge-
nous wage rigidity is non-linear, because the wage depends on the past wage Wt�1 and
the bounds of the bargaining setWUBt andWLB
t , multiplied by the degree of rigidity Rt,
which itself depends again on Wt and WUBt and WLB
t , see (26) and (27). If we were to
log-linearize this wage rule, it would reduce to a partial adjustment rule with a constant
degree of wage rigidity. Therefore, we use a second-order approximation of the policy
functions. As an accuracy check, Figure 3 shows that a second-order approximation
captures well the non-linear wage rule for � = 1, but for � = 2 or larger, a higher-order
approximation is needed. We simulate the second-order approximation of the model
201; 000 periods, discarding the �rst 1; 000 observations to eliminate the e¤ect of the
initial conditions. The results of this exercise are reported in Table 7.
4.5.1 Flexible Wages
Labor productivity is strongly procyclical in terms of its correlation with output in
the model with a frictional labor market, and its procyclicality falls substantially as
we reduce labor market frictions. The correlation of productivity with employment also
falls, from around zero in the frictional labor market to countercyclical in the frictionless
model. Both observations are consistent with the evidence. The reason for the decline in
22
the procyclicality of productivity, is the increase in the relative volatility of employment,
a result that is consistent with the data as well.
Two elements in the model are crucial for these results. First, the e¤ort choice
provides an intensive margin of adjustment for labor input. As frictions fall, it becomes
optimal to adjust labor more through employment and less through e¤ort. Thus, the
volatility of employment increases more than that of output. Second, �uctuations in the
model are driven by two types of shocks: technology shocks and preference shocks or
labor supply shocks. In a one-shock model, the correlations between all variables would
be close to either 1 or �1.30 In addition, if �uctuations were driven only by technology
shocks then productivity could never be countercyclical, since employment would only
�uctuate because of changes in labor demand, and the direct e¤ect of technology on
productivity would always prevail over the indirect e¤ect of employment. It is important
to stress, however, that our results are not driven by changes in the relative importance
of both shocks, which we keep constant, but by the reduction in frictions, which changes
the response of the economy conditional on each shock.
The model also predicts a small decrease in the relative volatility of wages and a
small increase in the volatility of output. As argued in section 2.3, the �rst prediction is
arguably not consistent with the data. The second one clearly is in contradiction with
the well-documented reduction in output volatility, the so called Great Moderation.
The decrease in the relative volatility of wages is driven by the fact that the wage
is approximately proportional to the marginal product of labor.31 Since the marginal
product of labor is proportional to output, but inversely proportional to employment,
an increase in the relative volatility of employment must necessarily also decrease the
relative volatility of wages. The increase in the volatility of output simply stems from the
fact that reducing adjustment costs ampli�es �uctuations in employment and therefore
in output as well. In the next subsection we show that endogneous wage rigidities
can easily reverse the predictions of the model for wages and possibly also bring the
prediction for output volatility closer to the evidence.
4.5.2 Endogenous Wage Rigidity
The third panel in Table 7 presents the simulated second moments for the model with
endogenous wage rigidity (� = 1 and �R = 0:95). Comparing these moments to those for
the �exible wage model, we see that the previously described predictions of the model
for the cyclicality of labor productivity and the relative volatility of employment remain
30This is exactly true in a static, linear model. Our model is close to (log)linear and the versionwithout capital and with �exible wages has only one state variable (employment), which has very fasttransition dynamics.31On a frictional labor market, the wage is not equal to marginal product of labor, but as long as
is not too large, they are still proportional since workers and �rms share the match surplus in equalproportions.
23
virtually unchanged. The reason is that the fact that wages adjust when they get close
to the bounds of the bargaining set mitigates the allocative e¤ect of wage rigidity.
The prediction of the model for the relative volatility of wages, however, is reversed.
The reduction in labor market frictions now increases the volatility of wages, although
not by as much as in the data. To understand the mechanism behind this result, it
is useful to consider the extreme case of endogenous wage rigidity (� ! 1), in which
wages are completely �xed within the bargaining set, but adjust when they hit the
bounds of the bargaining set. If frictions are high enough, so that the bargaining set
is very wide, wages never adjust and their volatility is zero. On the other extreme, on
a frictionless labor market, the bargaining set reduces to a point and wages behave as
if they were �exible. Of course this e¤ect is counteracted by the fact that the bounds
of the bargaining set themselves are less volatile when frictions are lower. However, for
our calibration of the parameters, the �rst e¤ect dominates, as illustrated in Figure 4.
The small increase in output volatility in response to the reduction in labor market
frictions is reversed to a small decrease for the model with endogenous wage rigidity.
The reason is that increased wage �exibility dampens �uctuations in output in response
to technology shocks.32 This result is consistent with a possible role of a decline in labor
market frictions as a source of the Great Moderation. However, the e¤ect is again much
smaller than in the data. Thus, whereas we �nd this last result intuitively compelling, it
is speculative in the sense that it is not clear whether it may be important quantitatively.
5 Conclusions
In this paper, we documented three changes in labor market dynamics over the postwar
period in the US: the strong procyclicality of labor productivity has vanished, the volatil-
ity of employment has increased with respect to output, and the volatility of wages has
increased relative to output and possibly even in absolute terms. We presented a model
to argue that a more �exible labor market, modelled as a reduction in hiring costs, can
explain all three facts. In addition, we showed that -in principle- the reduction in labor
market frictions may also have contributed to the reduction in output volatility, which
happened around the same time.
The intuition for why a reduction in labor market frictions increases the relative
volatility of employment and reduces the procyclicality of labor productivity is straight-
forward and compelling. If there is another input into production that can be used at
least partly as a substitute for labor, and which is not subject to the frictions, then a
reduction in frictions will make that input less volatile, so that employment becomes
more volatile with respect to output. In this paper, we refer to this other factor input
as e¤ort. But a very similar argument can be made for capacity utilization of capital.
32 In reponse to preference shocks, which a¤ect labor supply rather than labor demand, the reverse istrue. However, the e¤ect of technology shocks dominates that of preference shocks in our calibration.
24
Given that capital does not �uctuate much at business cycle frequencies, the fact that
the comovement of labor and output -and therefore labor productivity- has changed
almost unavoidably leads to the conclusion that there must be another input into the
production process.
Our argument that the reduction in labor market frictions is also responsible for the
increase in the relative volatility of wages is more contentious. Our simulations of the
model with endogenous wage ridigity show that this e¤ect is qualitatively present, and
can be made to dominate the direct e¤ect on wages for reasonable parameter values.
However, the increase in wage volatility in our simulations is much smaller than in the
data.33 Potentially, we can amplify this e¤ect by using a more non-linear wage rule,
but then we also need to use a higher-order approximation of the model. While there
is a compelling argument that wage rigidity is, at least to some degree, endogenous, it
remains an open question quantitatively how important this mechanism is.
What have we learned about the possible causes of the Great Moderation? We
showed that a reduction in labor market frictions, through an endogenous decrease in
wage rigidity, can lead to a decrease in output volatility. However, this e¤ect is small in
our simulations, partly because the decrease in wage rigidity is relatively small (at least
much smaller than in the data), but also because there is a direct e¤ect of the reduction
in labor market frictions that counteracts the e¤ect of decreased wage rigidity: reduced
frictions make labor more volatile, reducting the volatility of its marginal product. A
further caveat against the conclusion that the Great Moderation is driven by a reduction
in labor market frictions is that this story only works if technology shocks are relatively
important as a source of business cycle �uctuation. The reason is that preference shocks
act as labor supply shocks, so that wage rigidity dampens rather than ampli�es �uctu-
ations in employment in response to those shocks. Finally, reduced wage rigidity can
only lead to reduced output volatility if the wage is allocative, i.e. if the reduction in
wage rigidity applies to the wage of newly hired workers. We documented an increase
in the relative volatility of wages of new hires, see section 2.3.2, but that increase is
substantially smaller than for the average wage in the economy, casting further doubt
on the importance of this mechanism for the Great Moderation.
If it is true that the US labor market become more �exible in the mid 80s, then what
institutional change was responsible for it? It is tempting to attribute the lower labor
market frictions to improvements in recruitment technologies, particularly web-based
job search. However, that development, while potentially important, happened much
later.34 Major changes on the US labor market in the mid 80s include the introduction of
wrongful discharge laws in many states (Autor, Kerr, and Kugler (2007)), the increase
33At least than the data from the Labor Productivity and Cost program and the Current PopulationSurvey. The data from the Current Employment Statistics show a decrease in the relative volatility ofwages, see section 2.3, which our model has no trouble generating.34The largest and one of the �rst internet recruitment providers, monster.com, started in 1994.
25
in temporary help services and the decline of unionization. The introduction of the
wrongful discharge legislation constitutes an increase in employment protection, which
would increase rather than decrease frictions in our simple model. The increased share
of temporary help workers (workers employed by a temporary help agency rather than
directly by the employer where they work) is often seen as a response to increased
employment protection and happened very gradually, see Estevão and Lach (1999),
whereas the changes we document seem to have happened relatively suddenly around
1984.
In terms of timing, the decline in union power lines up very well with our story.
Farber and Western (2002) document a sharp decline in the number of certi�cation
elections in the early 80s and interpret this as evidence for an �unfavourable political
climate which raises the costs of unionization�, induced by the Reagan�s policies and
in particular his handling of the air-tra¢c controllers� strike in 1981. Moreover, recent
evidence suggests that union power is an important institution to explain di¤erences in
labor market volatility across countries and time periods, see Kent, Smith, and Holloway
(2005), Bertola and LoPrete (2009) and Gnocchi and Pappa (2009). A logical next step
for future research would be to write down a model with unions and endogenize the
reduction in labor market frictions.
26
A Marginal Product and Disutility of E¤ort
This appendix derives the marginal product of employment to the �rm, equation (12),
and the marginal disutility from employment, expressed in consumption terms, to the
household, equation (16), if e¤ort adjusts endogenously. From equations (4) and (2), it
is straightforward di¤erentation to decompose the total e¤ect of employment on output
and total e¤ective labor supply into a direct e¤ect and an e¤ect through the endogenous
response of e¤ort.
dYjtdNjt
=@Yjt@Njt
+@Yjt@Ejt
@Ejt@Njt
=(1� �)Yjt
Njt
�
1 + Njt
Ejt
@Ejt@Njt
�
(48)
dLhtdNht
=@Lht@Nht
+@Lht@Eht
@Eht@Nht
=1
1 + �
�
1 + �E1+�ht
�
1 + (1 + �)Nht
Eht
@Eht@Nht
��
(49)
Here, Ejt denotes the e¤ort of all workers i that are employed in �rm j and Eht the e¤ort
of all workers that are members of household h.
To �nd the response of e¤ort to changes in employment that �rm and household
face, we use the condition that the marginal disutility from e¤ort of a given worker
i (expressed in consumption terms) from equation (8), in equilibrium must equal the
marginal productivity of that worker to the �rm from equation (9).
E1+�� it =
(1 + �)
(1 + �) �
Zt C�ht
(1� �)At
�Z Njt
0E vtdv
���
(50)
First, suppose �rm j considers employing Njt workers, given that all other �rms
employ the equilibrium number of workers Nt. Because there are in�nitely many �rms,
�rm j�s decision to employ Njt 66= Nt workers does not a¤ect the fraction of household h�s
members that are employed, so that by the assumption of perfect risk-sharing within the
household, the consumption of workers in �rm j is not a¤ected, Cht = Ct. Substituting
this, as well as the condition that all workers in �rm j exert the same amount of e¤ort,
Eit = Ejt for all i 2 [0; Njt], the e¤ort condition becomes,
E1+�� jt =
(1 + �)
(1 + �) �
Zt C�t
(1� �)At
�
E jtNjt
�
��
(51)
so that the elasticity of e¤ort in a given �rm j with respect to employment in that �rm,
is given byNjt
Ejt
@Ejt@Njt
= ��
1 + �� (1� �) (52)
Substituting this elasticity into equation (48) above, gives expression (12) in the text.
Next, suppose household h considers having Nht employed workers, given that all
other households have Nt employed workers. Because there are in�nitely many house-
holds, household�s h�s decision to have a fraction of Nht 66= Nt of its members employed,
27
does not a¤ect the level of employment in any �rm Njt = Nt. Furthermore, although
the e¤ort level of worker i may change because of household h�s decision, e¤ort of all
other workers in �rm j, who are members of di¤erent households, is una¤ected, Eit = Eht
and Ei0t = Et for i0 6= i. Thus, the e¤ort condition becomes,
E1+�� ht =
(1 + �)
(1 + �) �
Zt C�ht
(1� �)At
�
E t Nt
�
��
(53)
and the elasticity of e¤ort exerted by members of household h with respect to employ-
ment in that household, using equation (3), is given by,
Nht
Eht
@Eht@Nht
=ChtEht
@Eht@Cht
�Nht
Cht
@Cht@Nht
= ��
1 + ��
WhtNht
Cht= �
�
1 + �� (54)
Substituting this elasticity into equation (49) above, gives expression (16) in the text.
28
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Table 1. The Vanishing Procyclicality of Labor Productivity
Corr with output Corr with labor input
Pre-84 Post-84 Change Pre-84 Post-84 Change
Output per hour
BP 0:60 0:25 �0:35 0:19 �0:40 �0:59
[0:06] [0:07] [0:09] [0:09] [0:07] [0:11]
4D 0:62 0:18 �0:44 0:09 �0:54 �0:62
[0:05] [0:09] [0:10] [0:07] [0:08] [0:11]
Output per worker
BP 0:78 0:60 �0:18 0:31 �0:15 �0:47
[0:04] [0:05] [0:06] [0:08] [0:10] [0:13]
4D 0:76 0:46 �0:30 0:13 �0:33 �0:46
[0:03] [0:09] [0:09] [0:08] [0:11] [0:14]
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. Data for the private sector are from the BLS
labor productivity and cost program (LPC) and refer to the private sector. Labor input
is total hours worked in the �rst panel and employment in the second panel, consistent
with the de�nition of labor productivity. The sample period is 1949-2007.
33
Table 2. The Rising Volatility of Labor Input
Std. Dev. Relative Std. Dev.
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Employment (private sector)
BP 1:57 0:91 0:58 0:66 0:81 1:23
[0:08] [0:05] [0:04] [0:03] [0:05] [0:09]
4D 2:44 1:54 0:63 0:66 0:94 1:43
[0:13] [0:12] [0:06] [0:03] [0:08] [0:15]
Hours (private sector)
BP 1:93 1:18 0:61 0:81 1:06 1:30
[0:09] [0:06] [0:04] [0:03] [0:05] [0:08]
4D 2:94 1:92 0:65 0:79 1:17 1:48
[0:15] [0:13] [0:06] [0:04] [0:09] [0:13]
Hours (total economy)
BP 1:68 0:85 0:51 0:71 0:76 1:07
[0:08] [0:04] [0:04] [0:03] [0:04] [0:07]
4D 2:56 1:47 0:57 0:69 0:89 1:30
[0:14] [0:10] [0:05] [0:03] [0:07] [0:11]
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. Data for the private sector are from the BLS
labor productivity and cost program (LPC) and refer to the private sector. Hours (total
economy) is an unpublished series for economy-wide hours constructed by the BLS and
used in Francis and Ramey (2008). The sample period is 1949-2007.
34
Table 3. The Rising Volatility of Wages
Std. Dev. Relative Std. Dev.
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Compensation per hour (NIPA, private sector)
BP 0:71 0:99 1:38 0:30 0:88 2:93
[0:05] [0:06] [0:12] [0:02] [0:07] [0:31]
4D 1:72 1:61 0:93 0:46 0:98 2:11
[0:12] [0:11] [0:09] [0:04] [0:12] [0:32]
Compensation per hour (NIPA, total economy)
BP 0:78 0:86 1:10 0:33 0:76 2:32
[0:05] [0:05] [0:10] [0:02] [0:07] [0:24]
4D 1:85 1:57 0:85 0:50 0:95 1:92
[0:10] [0:10] [0:07] [0:03] [0:12] [0:26]
Earnings per hour (CES, private sector)
BP 1:38 0:40 0:29 0:54 0:34 0:63
[0:12] [0:02] [0:03] [0:04] [0:03] [0:07]
4D 1:85 0:97 0:52 0:52 0:56 1:08
[0:14] [0:08] [0:06] [0:05] [0:06] [0:16]
Compensation per hour (NIPA, private sector, CPI de�ator)
BP 0:83 0:99 1:20 0:35 0:89 2:54
[0:05] [0:06] [0:10] [0:02] [0:08] [0:27]
4D 1:93 1:70 0:88 0:52 1:04 1:99
[0:13] [0:12] [0:08] [0:04] [0:13] [0:30]
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. Wages are calculated as real compensation
per hour. Compensation and hours data for the private sector are from the BLS labor
productivity and cost program. Compensation data for NIPA compensation are com-
bined with an unpublished economy-wide series for hours constructed by the BLS and
used in Francis and Ramey (2008). Compensation from the establishment survey or
Current Employment Statistics (CES) exclude non-wage payments. The sample period
is 1949-2007 for the NIPA and 1965-2004 for the CES-based series.
35
Table 4. The Rising Volatility of Wages: Newly Hired Workers
Std. Dev. Relative Std. Dev.
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Compensation per hour (NIPA, private sector)
BP 0:46 1:01 2:20 0:18 0:88 4:78
[0:06] [0:06] [0:29] [0:04] [0:08] [1:01]
4D 1:71 1:63 0:95 0:37 0:97 2:62
[0:13] [0:11] [0:10] [0:05] [0:13] [0:50]
Mean usual hourly earnings (CPS)
BP 0:49 0:54 1:10 0:20 0:47 2:39
[0:05] [0:03] [0:13] [0:04] [0:04] [0:48]
4D 1:48 1:27 0:86 0:32 0:76 2:36
[0:14] [0:11] [0:11] [0:05] [0:09] [0:46]
Mean usual hourly earnings newly hired workers (CPS)
BP 1:42 0:67 0:47 0:57 0:58 1:03
[0:16] [0:05] [0:07] [0:08] [0:05] [0:17]
4D 4:49 2:77 0:62 0:97 1:65 1:69
[0:60] [0:26] [0:10] [0:17] [0:24] [0:38]
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. CPS wage data are earnings per hour from the
outgoing rotation groups, which limits the period for which quarterly data are available
to after 1979. Wage series are constructed as in Haefke, Sonntag, and van Rens (2008)
but are not corrected for composition bias for comparability with other data sources.
However, keeping the composition of the labor force contant in terms of education,
experience and demographic characteristics makes very little di¤ererence for the results
presented here. The sample period is 1980-2005.
36
Table 5. Additional Business Cycle Statistics
A. Volatility output and productivity
Std. Dev. Relative Std. Dev.
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Output
BP 2:37 1:12 0:47
[0:13] [0:06] [0:04]
4D 3:72 1:64 0:44
[0:18] [0:16] [0:05]
Output per worker
BP 1:36 0:81 0:60 0:57 0:72 1:26
[0:08] [0:05] [0:05] [0:03] [0:06] [0:12]
4D 2:50 1:26 0:51 0:67 0:77 1:15
[0:14] [0:08] [0:04] [0:03] [0:07] [0:13]
B. Correlations
Corr with output Corr with employment
Pre-84 Post-84 Change Pre-84 Post-84 Change
Employment (private sector)
BP 0:84 0:70 �0:14
[0:02] [0:05] [0:06]
4D 0:75 0:69 �0:06
[0:03] [0:05] [0:06]
Compensation per hour (NIPA, private sector)
BP 0:40 0:28 �0:12 0:20 �0:11 �0:31
[0:08] [0:09] [0:12] [0:09] [0:09] [0:13]
4D 0:30 0:11 �0:20 �0:02 �0:29 �0:27
[0:07] [0:07] [0:10] [0:07] [0:08] [0:11]
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. See tables 1, 2 and 3 for data sources. The
sample period is 1949-2007.
37
Table 6. Model Calibration
Parameter Target
Utility: � = 0:99 quarterly data
� = 1 log utility over consumption
= 1:24 frictionless employment population ratio �N = 0:7
Production: f (N) = N1��, � = 1=3 capital share
E¤ort: � = 0:299 normalization: frictionless E = 1
� = 0 normalization so that E is in utils
= 0:3 total curvature �+ is a free parameter
Frictions: � = 0:306 gross quarterly separations, Shimer (2007)
g (H) = 12�H
2 quadratic adjustment costs, Yashiv (2010)
� = 0� 1:35 frictions 0� 3% of output, Silva and Toledo (2009)
Shocks: �A = 0:97; �A = 0:186 normalization: sd(y) = 1%
�z = 0:97, �z = 0:173 sd(n) =sd(y) = 0:66
38
Table 7. Simulation results
empl/pop correlation productivity relative std.dev. std.dev.
ratio �N with output with empl empl nt wage wt output yt
Data
Pre-84 0:78 0:31 0:66 0:30
Post-84 0:60 �0:15 0:81 0:88
Model with �exible wages
frictions 3% 0:59 0:76 �0:05 0:66 0:89 1:00
frictions 2% 0:63 0:70 �0:15 0:72 0:88 0:99
frictions 1% 0:66 0:65 �0:25 0:78 0:89 1:00
frictionless 0:70 0:60 �0:31 0:84 0:86 1:02
Model with endogenous wage rigidity ( � = 1, �R = 0:95)
frictions 3% 0:59 0:74 0:13 0:66 0:71 1:00
frictions 2% 0:63 0:71 0:06 0:71 0:70 1:00
frictions 1% 0:66 0:64 �0:05 0:77 0:71 1:00
frictionless 0:70 0:66 �0:28 0:79 0:88 0:91
Moments for the model are based on simulated time series of 200; 000 quarters. We
simulate the model for 201; 000 quarters but ignore the �rst 1; 000 quarters to eliminate
the e¤ect of the initial conditions. Numbers in bold are calibration targets.
39
Figure 1. The Vanishing Procyclicality of Labor Productivity
-4-2
02
4
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Labor productivity (bandpass filter)
Output per hour in the US private sector. Shaded areas are NBER recessions.
Figure 2. The Vanishing Procyclicality of Labor Productivity: Rolling Correlations
-.5
0.5
1
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005
Correl prod with output (blue) and hours (red), cntrd 6-yr rolling window, bp
Correlations are calculated in a centered 6-year rolling window of quarterly bandpass-
�ltered data.
40
Figure 3. Endogenous Wage Rigidity: Wage Rule
� = 1 (quadratic)
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80.8
0.85
0.9
0.95
1
1.05
� = 2 (quartic)
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.80.8
0.85
0.9
0.95
1
1.05
Wage rigidity Rt as a function of the relative distance of the wage from the center of
the bargaining set (Wt �W�
t ) =12
�
WUBt �WLB
t
�
, see equation (23). The red diamonds
represent the theoretical non-linear relation. The blue circles are simulated data from a
second-order approximation of the model. In these graphs, frictions are set to 0:3% of
output.
41
Figure 4. Endogenous Wage Rigity: Simulated Wage Data
Large frictions (1% of output)
0 50 100 150 200 250 300 350 400 450 500
-0.55
-0.5
-0.45
-0.4
-0.35
-0.3
Small frictions (0:2% of output)
0 50 100 150 200 250 300 350 400 450 500
-0.55
-0.5
-0.45
-0.4
-0.35
-0.3
Simulated wage paths for the model with large frictions and small frictions. In this
calibration, the wage is more volatile if frictions are smaller.
42
B Additional tables and �gures (not for publication)
Extended Table 1. The Vanishing Procyclicality of Labor Productivity
Output per hour
Pre-84 Post-84 Change Pre-84 Post-84 Change Pre-84 Post-84 Change
1949 - 2007 BP 0.60 0.25 -0.35 0.09 -0.47 -0.56 0.19 -0.40 -0.59
[0.06] [0.07] [0.09] [0.09] [0.07] [0.12] [0.09] [0.07] [0.11]
HP 0.61 0.04 -0.57 0.07 -0.60 -0.67 0.17 -0.56 -0.73
[0.05] [0.09] [0.10] [0.08] [0.06] [0.10] [0.08] [0.07] [0.10]
4D 0.62 0.18 -0.44 -0.03 -0.53 -0.51 0.09 -0.54 -0.62
[0.05] [0.09] [0.10] [0.08] [0.08] [0.11] [0.07] [0.08] [0.11]
1965 - 2004 BP 0.65 0.29 -0.36 0.16 -0.44 -0.60 0.28 -0.36 -0.63
[0.07] [0.07] [0.10] [0.13] [0.08] [0.15] [0.12] [0.08] [0.15]
HP 0.64 0.04 -0.60 0.11 -0.59 -0.70 0.23 -0.54 -0.77
[0.06] [0.09] [0.11] [0.10] [0.07] [0.12] [0.10] [0.07] [0.12]
4D 0.62 0.12 -0.50 -0.01 -0.56 -0.55 0.11 -0.54 -0.65
[0.07] [0.09] [0.12] [0.10] [0.08] [0.13] [0.10] [0.09] [0.13]
1975 - 1994 BP 0.78 0.34 -0.45 0.52 -0.39 -0.92 0.61 -0.36 -0.98
[0.05] [0.09] [0.11] [0.09] [0.11] [0.15] [0.08] [0.11] [0.13]
HP 0.66 -0.01 -0.68 0.26 -0.65 -0.91 0.39 -0.60 -0.99
[0.08] [0.14] [0.17] [0.13] [0.08] [0.15] [0.12] [0.09] [0.15]
4D 0.54 0.20 -0.33 -0.03 -0.51 -0.48 0.08 -0.52 -0.60
[0.11] [0.11] [0.16] [0.14] [0.14] [0.20] [0.14] [0.14] [0.20]
Correlation with Output Correlation with Empl. Correlation with Hours
Output per worker
Pre-84 Post-84 Change Pre-84 Post-84 Change Pre-84 Post-84 Change
1949 - 2007 BP 0.78 0.60 -0.18 0.31 -0.15 -0.47 0.44 0.01 -0.43
[0.04] [0.05] [0.06] [0.08] [0.10] [0.13] [0.07] [0.10] [0.12]
HP 0.77 0.40 -0.37 0.25 -0.32 -0.57 0.40 -0.18 -0.58
[0.03] [0.08] [0.09] [0.07] [0.09] [0.11] [0.07] [0.10] [0.12]
4D 0.76 0.46 -0.30 0.13 -0.33 -0.46 0.30 -0.21 -0.51
[0.03] [0.09] [0.09] [0.08] [0.11] [0.14] [0.07] [0.12] [0.14]
1965 - 2004 BP 0.78 0.63 -0.15 0.33 -0.13 -0.46 0.47 0.06 -0.41
[0.05] [0.05] [0.07] [0.11] [0.10] [0.15] [0.10] [0.10] [0.14]
HP 0.76 0.41 -0.35 0.25 -0.31 -0.55 0.41 -0.15 -0.56
[0.04] [0.08] [0.09] [0.09] [0.09] [0.13] [0.08] [0.11] [0.14]
4D 0.75 0.43 -0.31 0.12 -0.33 -0.45 0.30 -0.20 -0.50
[0.04] [0.09] [0.10] [0.10] [0.12] [0.15] [0.09] [0.12] [0.15]
1975 - 1994 BP 0.86 0.70 -0.16 0.62 -0.03 -0.65 0.73 0.10 -0.63
[0.03] [0.05] [0.06] [0.07] [0.16] [0.18] [0.06] [0.15] [0.16]
HP 0.74 0.36 -0.38 0.35 -0.38 -0.73 0.50 -0.22 -0.72
[0.06] [0.13] [0.15] [0.11] [0.12] [0.16] [0.10] [0.15] [0.17]
4D 0.69 0.49 -0.19 0.13 -0.30 -0.43 0.28 -0.18 -0.46
[0.08] [0.12] [0.14] [0.13] [0.19] [0.23] [0.12] [0.19] [0.23]
Correlation with HoursCorrelation with Output Correlation with Empl.
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. Data for the private sector are from the BLS
labor productivity and cost program (LPC) and refer to the private sector.
43
Extended Table 2. The Rising Volatility of Labor Input
Employment (private sector)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 1.57 0.91 0.58 0.66 0.81 1.23
[0.08] [0.05] [0.04] [0.03] [0.05] [0.09]
HP 1.62 1.16 0.72 0.66 0.97 1.47
[0.08] [0.07] [0.06] [0.03] [0.06] [0.12]
4D 2.44 1.54 0.63 0.66 0.94 1.43
[0.13] [0.12] [0.06] [0.03] [0.08] [0.15]
1975 - 1994 BP 1.71 0.83 0.49 0.65 0.71 1.11
[0.13] [0.07] [0.06] [0.04] [0.06] [0.11]
HP 2.10 1.27 0.60 0.71 1.01 1.41
[0.15] [0.11] [0.07] [0.05] [0.10] [0.17]
4D 2.94 1.60 0.54 0.73 0.91 1.24
[0.20] [0.13] [0.06] [0.06] [0.12] [0.20]
Std. Dev. Relative Std. Dev.
Hours (private sector)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 1.93 1.18 0.61 0.81 1.06 1.30
[0.09] [0.06] [0.04] [0.03] [0.05] [0.08]
HP 1.96 1.44 0.73 0.80 1.20 1.50
[0.10] [0.08] [0.05] [0.03] [0.07] [0.10]
4D 2.94 1.92 0.65 0.79 1.17 1.48
[0.15] [0.13] [0.06] [0.04] [0.09] [0.13]
1975 - 1994 BP 2.08 1.18 0.57 0.79 1.01 1.28
[0.15] [0.09] [0.06] [0.03] [0.07] [0.10]
HP 2.40 1.58 0.66 0.81 1.25 1.54
[0.19] [0.13] [0.08] [0.04] [0.11] [0.15]
4D 3.39 2.01 0.59 0.85 1.15 1.36
[0.27] [0.15] [0.07] [0.06] [0.14] [0.20]
Std. Dev. Relative Std. Dev.
Hours (total economy)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 1.68 0.85 0.51 0.71 0.76 1.07
[0.08] [0.04] [0.04] [0.03] [0.04] [0.07]
HP 1.71 1.06 0.62 0.70 0.89 1.27
[0.09] [0.07] [0.05] [0.03] [0.05] [0.09]
4D 2.56 1.47 0.57 0.69 0.89 1.30
[0.14] [0.10] [0.05] [0.03] [0.07] [0.11]
1975 - 1994 BP 1.63 0.94 0.58 0.62 0.81 1.31
[0.11] [0.06] [0.06] [0.03] [0.06] [0.11]
HP 1.98 1.22 0.61 0.67 0.97 1.44
[0.15] [0.10] [0.07] [0.03] [0.09] [0.15]
4D 2.67 1.54 0.58 0.67 0.88 1.32
[0.20] [0.11] [0.06] [0.05] [0.11] [0.19]
Std. Dev. Relative Std. Dev.
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. Data for the private sector are from the BLS
labor productivity and cost program (LPC) and refer to the private sector. Hours (total
economy) is an unpublished series for economy-wide hours constructed by the BLS and
used in Francis and Ramey (2008).
44
Extended Table 3. The Rising Volatility of Wages
Compensation per hour (NIPA, private sector)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 0.71 0.99 1.38 0.30 0.88 2.93
[0.05] [0.06] [0.12] [0.02] [0.07] [0.31]
HP 0.85 1.03 1.21 0.35 0.86 2.48
[0.06] [0.06] [0.11] [0.03] [0.08] [0.29]
4D 1.72 1.61 0.93 0.46 0.98 2.11
[0.12] [0.11] [0.09] [0.04] [0.12] [0.32]
1965 - 2004 BP 0.73 1.03 1.42 0.29 0.89 3.09
[0.07] [0.06] [0.16] [0.02] [0.08] [0.35]
HP 0.80 1.07 1.35 0.31 0.86 2.80
[0.06] [0.07] [0.14] [0.03] [0.08] [0.37]
4D 1.43 1.64 1.15 0.40 0.96 2.39
[0.10] [0.12] [0.11] [0.03] [0.13] [0.38]
1975 - 1994 BP 0.65 1.22 1.86 0.25 1.04 4.22
[0.07] [0.08] [0.24] [0.02] [0.13] [0.67]
HP 0.76 1.12 1.46 0.26 0.89 3.43
[0.08] [0.09] [0.18] [0.03] [0.10] [0.60]
4D 1.42 1.75 1.24 0.35 1.00 2.82
[0.11] [0.17] [0.15] [0.04] [0.21] [0.68]
Std. Dev. Relative Std. Dev.
Compensation per hour (NIPA, total economy)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 0.78 0.86 1.10 0.33 0.76 2.32
[0.05] [0.05] [0.10] [0.02] [0.07] [0.24]
HP 0.84 0.95 1.14 0.34 0.80 2.33
[0.05] [0.08] [0.11] [0.02] [0.09] [0.30]
4D 1.85 1.57 0.85 0.50 0.95 1.92
[0.10] [0.10] [0.07] [0.03] [0.12] [0.26]
1965 - 2004 BP 0.84 0.91 1.09 0.33 0.78 2.36
[0.08] [0.05] [0.12] [0.02] [0.07] [0.27]
HP 0.86 0.99 1.14 0.33 0.79 2.37
[0.08] [0.08] [0.14] [0.03] [0.09] [0.35]
4D 1.67 1.59 0.95 0.47 0.92 1.98
[0.11] [0.11] [0.09] [0.04] [0.12] [0.30]
1975 - 1994 BP 0.78 1.08 1.39 0.29 0.93 3.15
[0.10] [0.07] [0.20] [0.03] [0.11] [0.52]
HP 0.81 1.09 1.34 0.28 0.86 3.14
[0.08] [0.13] [0.20] [0.03] [0.14] [0.62]
4D 1.68 1.86 1.11 0.42 1.06 2.53
[0.14] [0.17] [0.14] [0.05] [0.21] [0.58]
Std. Dev. Relative Std. Dev.
continued on next page ...
45
Earnings per hour (CES, private sector)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1965 - 2004 BP 1.38 0.40 0.29 0.54 0.34 0.63
[0.12] [0.02] [0.03] [0.04] [0.03] [0.07]
HP 1.27 0.46 0.36 0.49 0.37 0.75
[0.11] [0.03] [0.04] [0.05] [0.04] [0.10]
4D 1.85 0.97 0.52 0.52 0.56 1.08
[0.14] [0.08] [0.06] [0.05] [0.06] [0.16]
1975 - 1994 BP 1.32 0.36 0.27 0.50 0.30 0.61
[0.15] [0.04] [0.04] [0.05] [0.04] [0.11]
HP 1.21 0.35 0.29 0.41 0.27 0.67
[0.14] [0.05] [0.05] [0.05] [0.04] [0.13]
4D 1.63 0.82 0.50 0.41 0.47 1.14
[0.19] [0.10] [0.08] [0.05] [0.07] [0.23]
Std. Dev. Relative Std. Dev.
Compensation per hour (NIPA, private sector, output de�ator)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 0.67 0.95 1.42 0.28 0.85 3.02
[0.04] [0.07] [0.13] [0.02] [0.08] [0.34]
HP 0.82 1.06 1.30 0.33 0.89 2.65
[0.06] [0.07] [0.12] [0.03] [0.08] [0.32]
4D 1.48 1.65 1.11 0.40 1.00 2.51
[0.11] [0.11] [0.11] [0.04] [0.13] [0.39]
1975 - 1994 BP 0.62 1.22 1.97 0.23 1.04 4.47
[0.08] [0.10] [0.29] [0.03] [0.14] [0.80]
HP 0.70 1.10 1.57 0.24 0.88 3.68
[0.09] [0.10] [0.23] [0.04] [0.11] [0.70]
4D 1.26 1.69 1.34 0.32 0.96 3.05
[0.09] [0.18] [0.17] [0.04] [0.21] [0.74]
Std. Dev. Relative Std. Dev.
Compensation per hour (NIPA, private sector, CPI de�ator)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1949 - 2007 BP 0.83 0.99 1.20 0.35 0.89 2.54
[0.05] [0.06] [0.10] [0.02] [0.08] [0.27]
HP 0.96 1.04 1.08 0.39 0.87 2.21
[0.06] [0.06] [0.09] [0.03] [0.08] [0.26]
4D 1.93 1.70 0.88 0.52 1.04 1.99
[0.13] [0.12] [0.08] [0.04] [0.13] [0.30]
1975 - 1994 BP 1.00 1.19 1.19 0.38 1.02 2.68
[0.09] [0.09] [0.13] [0.04] [0.13] [0.44]
HP 1.16 1.14 0.98 0.39 0.90 2.30
[0.09] [0.09] [0.11] [0.05] [0.11] [0.40]
4D 2.01 1.94 0.96 0.50 1.10 2.20
[0.24] [0.19] [0.15] [0.07] [0.23] [0.56]
Std. Dev. Relative Std. Dev.
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. Wages are calculated as real compensation per
hour. Compensation and hours data for the private sector are from the BLS labor pro-
ductivity and cost program. Compensation data for NIPA compensation are combined
with an unpublished economy-wide series for hours constructed by the BLS and used
in Francis and Ramey (2008). Compensation from the establishment survey or Current
Employment Statistics (CES) exclude non-wage payments.
46
Extended Table 4. The Rising Volatility of Wages: Newly Hired Workers
Compensation per hour (NIPA, private sector)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
1980 - 2005 BP 0.46 1.01 2.20 0.18 0.88 4.78
[0.06] [0.06] [0.29] [0.04] [0.08] [1.01]
HP 0.59 1.05 1.78 0.20 0.86 4.22
[0.08] [0.07] [0.27] [0.05] [0.08] [1.09]
4D 1.71 1.63 0.95 0.37 0.97 2.62
[0.13] [0.11] [0.10] [0.05] [0.13] [0.50]
Std. Dev. Relative Std. Dev.
Usual hourly earnings (CPS)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Median BP 0.54 0.93 1.73 0.22 0.81 3.77
[0.08] [0.08] [0.28] [0.03] [0.08] [0.60]
HP 0.78 1.14 1.46 0.27 0.93 3.46
[0.14] [0.10] [0.28] [0.06] [0.10] [0.84]
4D 1.59 1.75 1.10 0.35 1.04 3.02
[0.28] [0.12] [0.21] [0.07] [0.08] [0.65]
Mean BP 0.49 0.54 1.10 0.20 0.47 2.39
[0.05] [0.03] [0.13] [0.04] [0.04] [0.48]
HP 0.66 0.83 1.27 0.23 0.68 3.02
[0.11] [0.06] [0.24] [0.05] [0.07] [0.78]
4D 1.48 1.27 0.86 0.32 0.76 2.36
[0.14] [0.11] [0.11] [0.05] [0.09] [0.46]
Std. Dev. Relative Std. Dev.
Usual hourly earnings newly hired workers (CPS)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Median BP 1.96 1.09 0.56 0.79 0.95 1.21
[0.25] [0.08] [0.08] [0.09] [0.10] [0.19]
HP 5.08 2.94 0.58 1.75 2.40 1.37
[0.73] [0.24] [0.10] [0.35] [0.25] [0.31]
4D 5.67 3.40 0.60 1.23 2.02 1.65
[0.63] [0.27] [0.08] [0.18] [0.25] [0.31]
Mean BP 1.42 0.67 0.47 0.57 0.58 1.03
[0.16] [0.05] [0.07] [0.08] [0.05] [0.17]
HP 3.54 2.31 0.65 1.22 1.88 1.54
[0.47] [0.18] [0.10] [0.25] [0.18] [0.35]
4D 4.49 2.77 0.62 0.97 1.65 1.69
[0.60] [0.26] [0.10] [0.17] [0.24] [0.38]
Std. Dev. Relative Std. Dev.
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. CPS wage data are earnings per hour from the
outgoing rotation groups, which limits the period for which quarterly data are available
to after 1980. Wage series are constructed as in Haefke, Sonntag, and van Rens (2008)
but are not corrected for composition bias for comparability with other data sources.
However, keeping the composition of the labor force contant in terms of education,
experience and demographic characteristics makes very little di¤ererence for the results
presented here.
47
Extended Table 5. Additional Business Cycle Statistics
Volatility output and productivity (1949-2007)
Pre-84 Post-84 Ratio Pre-84 Post-84 Ratio
Output BP 2.37 1.12 0.47
[0.13] [0.06] [0.04]
HP 2.45 1.20 0.49
[0.13] [0.07] [0.04]
4D 3.72 1.64 0.44
[0.18] [0.16] [0.05]
/worker BP 1.36 0.81 0.60 0.57 0.72 1.26
[0.08] [0.05] [0.05] [0.03] [0.06] [0.12]
HP 1.48 0.85 0.58 0.60 0.71 1.18
[0.08] [0.06] [0.05] [0.03] [0.06] [0.12]
4D 2.50 1.26 0.51 0.67 0.77 1.15
[0.14] [0.08] [0.04] [0.03] [0.07] [0.13]
/hour BP 1.06 0.75 0.71 0.45 0.67 1.51
[0.06] [0.06] [0.07] [0.03] [0.07] [0.18]
HP 1.17 0.85 0.72 0.48 0.71 1.48
[0.07] [0.06] [0.07] [0.03] [0.07] [0.18]
4D 2.04 1.32 0.64 0.55 0.80 1.46
[0.13] [0.08] [0.06] [0.03] [0.09] [0.19]
Std. Dev. Relative Std. Dev.
Correlations (1949-2007)
Pre-84 Post-84 Change Pre-84 Post-84 Change Pre-84 Post-84 Change
Employment BP 0.84 0.70 -0.14
[0.02] [0.05] [0.06]
HP 0.81 0.74 -0.07
[0.03] [0.04] [0.05]
4D 0.75 0.69 -0.06
[0.03] [0.05] [0.06]
Hours BP 0.90 0.79 -0.11
[0.02] [0.04] [0.04]
HP 0.88 0.81 -0.07
[0.02] [0.03] [0.04]
4D 0.84 0.74 -0.10
[0.02] [0.05] [0.05]
Wage BP 0.40 0.28 -0.12 0.20 -0.11 -0.31 0.22 -0.09 -0.31
[0.08] [0.09] [0.12] [0.09] [0.09] [0.13] [0.09] [0.10] [0.14]
HP 0.37 -0.01 -0.38 0.23 -0.31 -0.54 0.21 -0.31 -0.52
[0.08] [0.10] [0.12] [0.09] [0.10] [0.14] [0.09] [0.10] [0.14]
4D 0.30 0.11 -0.20 -0.02 -0.29 -0.27 0.01 -0.36 -0.36
[0.07] [0.07] [0.10] [0.07] [0.08] [0.11] [0.07] [0.08] [0.11]
Correlation with Output Correlation with Empl. Correlation with Hours
Standard errors in brackets are calculated from the variance-covariance matrix of the
second moments using the delta method. See tables 1, 2 and 3 for data sources.
48
Figure 5. The Rise of Temporary Help Services
Source: Estevão and Lach (1999)
Figure 6. The Decline of Unions
Source: Farber and Western (2002)
49